2020-10-01T15:27:20Z
Acceptance in OA@INAF
Mass-Richness Relations for X-Ray and SZE-selected Clusters at 0.4 < z < 2.0 as
þÿSeen by Spitzer at 4.5 ¼m
Title
Rettura, A.; Chary, R.; Krick, J.; ETTORI, STEFANO
Authors
10.3847/1538-4357/aad818
DOI
http://hdl.handle.net/20.500.12386/27537
Handle
THE ASTROPHYSICAL JOURNAL
Journal
867
Mass
–Richness Relations for X-Ray and SZE-selected Clusters at 0.4<z<2.0
as Seen by
Spitzer at 4.5 μm
A. Rettura1,2 , R. Chary2 , J. Krick2, and S. Ettori3 1
W. M. Keck Observatory, Kamuela, HI 96743, USA
2
IPAC, Caltech, KS 314-6, Pasadena, CA 91125, USA
3
INAF, Osservatorio Astronomico di Bologna, via Ranzani 1, I-40127 Bologna, Italy Received 2016 August 31; revised 2018 July 31; accepted 2018 August 1; published 2018 October 25
Abstract
We study the mass–richness relation of 116 spectroscopically confirmed massive clusters at 0.4<z<2 by mining the Spitzer archive. We homogeneously measure the richness at 4.5μm for our cluster sample within a fixed aperture of 2′ radius and above a fixed brightness threshold, making appropriate corrections for both background galaxies and foreground stars. We have two subsamples, those which have (a) literature X-ray luminosities and(b) literature Sunyaev–Zel’dovich effect masses. For the X-ray subsample we re-derive masses adopting the most recent calibrations. We then calibrate an empirical mass–richness relation for the combined sample spanning more than one decade in cluster mass and find the associated uncertainties in mass at fixed richness to be±0.25 dex. We study the dependence of the scatter of this relation with galaxy concentration, defined as the ratio between richness measured within an aperture radius of 1 and 2 arcmin. Wefind that at fixed aperture radius the scatter increases for clusters with higher concentrations. We study the dependence of our richness estimates with depth of the 4.5μm imaging data and find that reaching a depth of at least [4.5]=21 AB mag is sufficient to derive reasonable mass estimates. We discuss the possible extension of our method to the mid-infrared WISE All Sky Survey data and the application of our results to the Euclid mission. This technique makes richness-based cluster mass estimates available for large samples of clusters at very low observational cost.
Key words: cosmology: observations– galaxies: clusters: general – galaxies: high-redshift – galaxies: statistics – infrared: galaxies– large-scale structure of universe
1. Introduction
Clusters of galaxies are the largest and most massive gravitationally bound systems in the universe. Clusters of galaxies are considered to be both unique astrophysical laboratories and powerful cosmological probes(e.g., White et al.1993; Bartlett & Silk1994; Viana & Liddle1999; Borgani et al.2001; Vikhlinin et al.2009; Mantz et al.2010; Rozo et al.2010; Allen et al.2011; Benson et al.2013; Bocquet et al.2015). Clusters grow from the highest density peaks in the early universe and thus their mass function is a tracer of the underlying cosmology (e.g., Press & Schechter1974; Bahcall & Cen1993; Gonzalez et al.2012). Due to the steep dependence between number density and mass in the dark-matter halo mass function, deriving the cluster mass accurately is of paramount importance and large observational efforts have been devoted to this goal over the past three decades. Different indirect methods, each of them leveraging unique observables of these systems, have been developed in the literature in order to weigh the most massive structures in the universe. These are(i) measuring the richness of a cluster, i.e., counting the number of galaxies associated with that cluster within a given radius(e.g., Abell1958; Zwicky & Kowal1968; Carlberg et al. 1996; Yee & López-Cruz 1999; Yee & Ellingson 2003; Rozo et al. 2009; Rykoff et al. 2012, 2014; Andreon & Congdon 2014; Andreon 2015, 2016; Saro et al. 2015; Melchior et al. 2017). (ii) Measuring the radial velocities of the cluster members, which yields the velocity dispersion of a cluster and can be used to derive the cluster’s mass from the virial theorem, under the assumption that the structure is virialized (e.g., Girardi et al. 1996; Mercurio et al. 2003; Demarco et al. 2005, 2007). (iii) Measuring the intensity of the hot X-ray-emitting intracluster medium if this
gas is in hydrostatic equilibrium by factoring in its density and temperature distribution (e.g., Gioia et al. 1990; Vikhlinin et al.1998; Böhringer et al.2000; Pacaud et al.2007;Šuhada et al. 2012; Ettori et al. 2013; Andreon et al. 2016). (iv) Measuring the inverse-Compton scatter of cosmic microwave background (CMB) photons off the energetic electrons in the hot intracluster gas. The resultant characteristic spectral distortion to the CMB is known as the Sunyaev–Zel’dovich effect (SZE; Sunyaev & Zeldovich 1972; Staniszewski et al.2009; Hasselfield et al. 2013; Bleem et al.2015; Planck Collaboration et al. 2015). (v) By measuring the coherent distortion that weak gravitational lensing produces on back-ground galaxies, which has the advantage that it does not need prior knowledge on the baryon fraction of the cluster or its dynamical state(e.g., Bartelmann & Schneider2001; Hoekstra 2007; Mahdavi et al. 2008; High et al. 2012; Hoekstra et al. 2012; Umetsu et al. 2014; von der Linden et al. 2014; Sereno2015).
While there are large cluster samples selected from optical and near-infrared photometric surveys up to z<1.5 (e.g., Gladders & Yee2000; Koester et al. 2007; Hao et al. 2010; Menanteau et al.2010; Brodwin et al.2011; Wen et al.2012; Ascaso et al.2014; Rykoff et al.2014; Bleem et al.2015), in recent years, mid-infrared (MIR) photometric surveys with Spitzer have extended the landscape. The Infrared Array Camera (IRAC; Fazio et al. 2004) onboard the Spitzer Space Telescope has proven to be a sensitive tool for studying galaxy clusters. Ongoing Spitzer wide-area surveys are proving effective at identifying large samples of galaxy clusters down to low masses at 1.5<z<2 (e.g., SDWFS, SWIRE, CARLA, SSDF; Eisenhardt et al.2008; Papovich2008; Wilson et al. 2009; Demarco et al. 2010; Galametz et al. 2010; © 2018. The American Astronomical Society. All rights reserved.
Stanford et al.2012; Zeimann et al.2012; Brodwin et al.2013; Galametz et al. 2013; Muzzin et al. 2013; Wylezalek et al. 2013; Rettura et al. 2014), where current X-ray and SZE observations are restricted to only the most massive systems at these redshifts (Brodwin et al. 2011; Muzzin et al.2013).
Even larger samples of clusters at 0.4<z<2.0 will soon be available from upcoming and planned large-scale surveys like the Dark Energy Survey (DES; DES Collaboration et al. 2018), KiDS (de Jong et al.2013), Euclid (Laureijs et al.2011), LSST(LSST Science Collaboration et al.2009), and WFIRST. However, until the next generation SZE instrumentation (e.g., ACTpol, SPTpol, SPT3G—in any case only covering the southern sky) or next generation X-ray telescopes (e.g., eRosita, Athena) become available, measuring the masses of the bulk of the high-redshift clusters at 0.4<z2 remains challenging.
In order to provide an efficient and reliable mass proxy for high-redshift clusters up to z∼2, in this paper, we calibrate a richness–mass relation using archival 4.5 μm data on a sample of published X-ray and SZE-selected clusters at 0.4< z<2.0. At these redshifts, the 4.5 μm band traces rest-frame near-infrared light from the galaxies that is emitted by the high mass-to-light ratio stellar population. Thus, if the integrated mass function of galaxies is correlated with the cluster dark-matter halo in which they reside, the near-infrared richness should provide a reasonable tracer of cluster mass (e.g., Andreon 2006, 2013). This method of mass measurement has the advantage over the others described above because it is purely photometric, does not require a priori knowledge of the dynamical state of the cluster, and is observationally easy to obtain. We require only the cluster position, an approximate redshift estimate and at least 90 s depth coverage of IRAC 4.5μm data over a single pointing of 5′×5′ field of view.
The plan of the paper is as follows. In Section2, we describe the archival cluster sample we have adopted throughout the work and describe how the cluster masses were derived. In Section 3, we present the Spitzer photometric cataloging procedure adopted. In Section 4, we present the definition of our richness indicator and study its dependence on survey depth and aperture radius adopted. In Section 5, we calibrate the mass–richness relation for each subsample individually and combined. In Section6, we discuss our results, the possibility of extending our method to other MIR all-sky surveys, and the implication of our findings on future wide-field infrared surveys such as those that will be undertaken with Euclid. In Section 7, we summarize the results.
Throughout, we adopt a ΩΛ=0.7, Ωm=0.3, and H0=
70 km s−1Mpc−1 cosmology, and use magnitudes in the AB system.
2. Sample Selection
In the following section we present cluster samples drawn from the literature and the archival Spitzer data adopted in our analysis. Our aim is to assemble a large sample of clusters with known masses and redshifts for which archival IRAC data at 4.5μm is publicly available. We define two cluster subsamples based on literature X-ray masses and literature SZE masses.
2.1. X-Ray Clusters Sample
The starting point for this sample is the Meta-catalog of X-ray detected clusters of galaxies (MCXC), a catalog of compiled properties of X-ray detected clusters of galaxies (Piffaretti et al. 2011, and references therein). This catalog is based on the ROSAT All Sky Survey(Voges et al.1999) data on 1743 clusters at 0.003<z<1.261 that have been homogeneously evaluated within the radius, R500,
corresp-onding to an overdensity of 500 times the critical density. For each cluster, the MCXC provides redshift,4coordinates, R500,
and X-ray luminosity in the 0.1–2.4 keV band, L500,[0.1–2.4keV]. Based on the values published in Piffaretti et al.(2011), we also derive the angular size, θ500=R500/DA(z) for each cluster,
where DA(z) is the angular diameter distance.
In order to define a richness parameter to be used as a proxy for cluster mass, M500, we need to define the aperture radius in
which galaxies should be counted and the redshift range for which this radius is still representative of the cluster R500. To
this aim, in Figure1, we show the entire MCXC sample θ500
versus redshift relation. The red horizontal line indicates the 2.5 arcmin radius of a single Spitzer/IRAC field of view. The red asterisks indicate the mean θ500 per redshift bin of
Δz=0.1, the error bars are the standard deviation of the mean per redshift bin. We note that at z>0.4 (dot–dashed line) the averageθ500of the sample is included within the Spitzer/IRAC
field of view. Therefore we adopt this lower redshift cut to the cluster samples considered in our study. There are 142 clusters in MCXC at z>0.4.
The most reliable X-ray masses are obtained by solving the equation of hydrostatic equilibrium, which requires measure-ments of the density and temperature gradients of the X-ray emitting gas (see discussion in Maughan 2007). This is only possible for nearby bright clusters, therefore it remains a challenge for the majority of clusters detected in X-ray surveys, especially at high redshifts where surface brightness dimming effects become significant. Thus, in most cases, cluster masses are estimated from simple properties such as X-ray luminosities
Figure 1.θ500as a function of redshift for the entire MCXC sample of X-ray clusters. The horizontal red line indicates half the typicalfield of view of a single Spitzer/IRAC pointing. At z>0.4 (dot–dashed line), the average θ500 of the sample is included in the Spitzer/IRAC field of view. Asterisks are average values in bins of redshift of sizeΔz=0.1. The final X-ray sample analyzed in this study is indicated with solid red circles.
4
(LX) or from adopting a single global temperature (kT) via the
calibration of scaling relations.
To derive an estimate of the total cluster mass, M500, within
R500, we adopt the most recent calibrations, in particular in their
redshift evolution, of the relations between X-ray global properties and cluster total mass, as presented in Reichert et al.(2011). Reichert et al. (2011, see also references therein) obtained these relations by homogenizing published estimates of X-ray luminosity and total mass. These values were rescaled at different radii and overdensities by using their dependence upon the gas density, which was described by a β-model (Cavaliere & Fusco-Femiano1976).
Reichert et al. (2011) scaling relations, together with the MCXC luminosities, are then used here to run the following iterative process: (i) an input temperature is assumed; (ii) a conversion from the MCXC L500,[0.1–2.4keV] luminosity to the
pseudo-bolometric (0.01–100 keV) value, L500,bol, is derived
assuming the thermalapec model in XSPEC (Arnaud 1996), adopting the temperature assumed at step (i) and a metal abundance of 0.3 times the solar value;(iii) a value of the mass within an overdensity of 500 with respect to the critical density of the universe at the cluster’s redshift is then calculated from Equation(26) in Reichert et al. (2011),
= a - ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ⎛ ⎝ ⎜ ⎞⎠⎟ ( ) · · ( ) ( ) M M L H z H 10 1.64 0.07 10 erg s , 1 500,X 14 500,bol 44 1 0.52 0.03 0 whereH z( ) = W + WL *m(1+z)3+ W*k(1+z)2,Ωk=(1 − Ωm− ΩΛ), and a = -0.90-+0.150.35;
(iv) a new temperature is recovered from the M–T relation (Equation (23) in Reichert et al. 2011) and compared to the input value assumed at step(i); (v) the calculations are repeated if the relative difference between these two values is larger than 5%.
We consider also a correction on the given luminosity due to the change in the initial R500. This correction, typically a few
percent, is obtained as described in Piffaretti et al.(2011), by evaluating the relative change of the square of the gas density profile integrated over the cylinder with dimension of r=R500
and height of 2×5 R500.
As a consistency test, for a subsample of common clusters, we can also compare the bolometric luminosities we have obtained with those independently derived by Maughan et al. (2012). Maughan et al. (2012) used a sample of 115 galaxy clusters at 0.1<z<1.3 observed with Chandra to investigate the relation between X-ray bolometric luminosity and YX(the
product of gas mass and temperature) and found a tight LX–YX
relation (Maughan 2007). They also demonstrate that cluster masses can be reliably estimated from simple luminosity measurements in low quality data where direct masses, or measurements of YX, are not possible.
There are 26 clusters in common between our ROSAT-based sample and their Chandra sample. In Figure2,we compare the bolometric luminosities obtained independently and find the values to be in very good agreement.
We then searched for Spitzer/IRAC archival observations homogeneously covering at least an area within a 2.5 arcmin radius from the cluster center coordinates and with a minimum exposure time of 90 s. This depth ensures that we reach at least
a 5σ sensitivity limit of 21.46 AB mag (9.4 μJy) at 4.5 μm (see Section3.1for further discussion of required depth).
These requirements result in a final X-ray-selected sample comprised of 47 galaxy clusters at 0.4<z<1.27 (indicated by red circles in Figure1). We note that a few large clusters (indicated by red circles above the red line in Figure 1) have still been considered throughout this work. This is because the meanθ500in those redshift bins is smaller than the IRACfield
of view. It also ensures an adequate sample size and avoids biasing our derived richness–mass relation against large, less-concentrated clusters. The derived cluster mass and redshift distributions of our X-ray sample are illustrated in Figure 3 (blue circles and histograms).
2.2. SZE Clusters Sample
Recent years have seen rapid progress of both the quality and quantity of SZE measurements using a variety of instruments. Therefore, several programs have been launched in the past few years with the aim of measuring total masses through the SZ effect of large samples of clusters, both for cosmology and astrophysics studies. Spitzer/IRAC data coverage over some of the SZ surveyfields have been requested, as well as targeted SZE observations of existing MIR-selected clusters have also been obtained by various investigators.
We have therefore mined the Spitzer/IRAC archive and drawn a heterogeneous sample of spectroscopically confirmed SZE-selected clusters based from a number of these programs. Applying the same redshift and photometric coverage selection criteria illustrated in Section2.1, thefinal SZE-selected sample considered in our study is comprised of 69 galaxy clusters at 0.4<z<2.0.
In particular, our sample is comprised of 4 clusters from the Planck Cluster Catalog (Planck Collaboration et al. 2015), 4 clusters from the Massive Distant Clusters of WISE Survey (MADCoWS, Brodwin et al. 2015), 1 cluster from the IRAC
Figure 2. ROSAT-based bolometric luminosities derived in this work are plotted against those measured with Chandra by Maughan et al.(2012) for a
Distant Cluster Survey (IDCS, Brodwin et al.2012), 1 cluster from the XMM-Newton Large Scale Structure Survey(XLSSU, Pierre et al.2011; Mantz et al.2014), and 59 clusters from the SPT-SZ Cluster Survey (SPT-SZ, Bleem et al.2015),.
Cluster masses, M500,SZ, as reported in the aforementioned
papers, are based on the spherically integrated Comptonization measurement, Y500,SZ, obtained by either the Planck Space
Telescope, the Combined Array for Research in Millimeter-wave Astronomy (CARMA5), or the South Pole Telescope (SPT; Carlstrom et al. 2011; Austermann et al. 2012; Story et al.2013). Cluster mass and redshift distributions of the final SZE-selected subsample are illustrated in Figure 3(red circles and histograms) and can be compared with the X-ray sample shown therein.
We also note that onlyfive clusters in our sample, Clus ID 26, 355, 621, 1050, and OBJ8 have mass estimates in the literature, derived both from LXand Y500,SZ, and that with the
exception of Clus ID 621, 1050, the majority have consistent mass estimates within 2σ of the associated errors.
3.Spitzer Data
Publicly available Spitzer/IRAC data for each cluster in our sample is accessible via the Spitzer Heritage Archive (SHA). All of the IRAC data for the X-ray-selected sample were acquired during the initial cryogenic mission, while all but four of the SZE sample data were acquired during the post-cryogenic Warm Mission. The Warm and Cryo missions have been put onto the same calibration scale in the SHA provided data products, so we expect no differences between the missions to be relevant to this work.
3.1. Source Extraction
The publicly accessible Spitzer Enhanced Imaging Products6 (SEIP) provide super mosaics (combining data from multiple programs where available) and a source list of photometry for sources observed during the cryogenic mission of Spitzer. The SEIP includes data from the four channels of IRAC (3.6, 4.5, 5.8, 8μm) and the 24 μm channel of the Multi-Band Imaging Photometer for Spitzer(MIPS) where available. In addition to the Spitzer photometry, the source list also contains photometry for positional counterparts found in the AllWISE release of the Wide-Field Infrared Survey Explorer (WISE) and in the Two Micron All Sky Survey(2MASS). To ensure high reliability, strict cuts were placed on extracted sources and some legitimate sources may appear to be missing. These sources were removed by cuts in size, compactness, blending, shape, and SNR, along with multiband detection requirements. In mostfields, the completeness of the source list is well matched to expectations for a SNR=10 cutoff, as reliability is favored over completeness. However, the list may be incomplete in areas of high surface brightness and/or high source surface density. This is most relevant for this work for objects near bright sources or the centers of clusters, which may have a higher source density.
Following the recommendations in Surace et al.(2004), for our richness estimate we adopt the aperture corrected IRAC 4.5μm flux density measured within an aperture of diameter 3.8 arcsec from the SEIP source list. The chosen aperture is twice the instrumental FWHM, which provides accurate photometry with an aperture correction for a point source already applied, which is customary for cluster studies with Spitzer in the literature(e.g., Bremer et al.2006; Rettura et al. 2006). IRAC PSF has a FWHM∼2 arcsec, thus we note that a star/galaxy separation in Spitzer data, especially at faint fluxes, is not straightforward. Therefore we will describe in Section4 how we account and correct for foreground stars in our richness estimates.
For the Warm Mission data, a SEIP source list is not available in the Spitzer archive. However, we have adopted the same SEIP source extraction pipeline and applied it ourselves in exactly the same way as for the Cryo mission clusters.
3.2. Survey Depth
As we deal with a heterogeneous sample that has been observed by Spitzer at varying depths, for consistency of our analysis, we aim to be able to calibrate our method to a depth that is reached by all our archival data.
For illustration purposes, we show in Figure4 the number counts of four representative clusters in our samples along with the number counts derived from a reference deep Spitzer legacy program that we adopt as a controlfield. The Spitzer UKIDSS Ultra Deep Survey (SpUDS, PI: J. Dunlop) data used here come from a program covering∼1 deg2in the UKIRT Infrared Deep Sky Survey, Ultra Deep Survey field (UKIDSS UDS Dye et al. 2006), centered at R.A.=02h:18m:45s, Decl.= −05°:00′:00″. Note that we use the SEIP source list photometry available in the archive for our control(SpUDS) field as well. The SpUDS survey reaches greater sensitivities than the data on the majority of our clusters, in particular for the SZE
Figure 3.Cluster mass and redshift distributions of the X-ray-(blue) and SZE-selected(red) cluster samples studied in this work. Both subsamples extend over similar ranges of the M500–z plane, and the median values are indicated by the dashed lines of the corresponding color.
5
sample, as shown by examples on the right column of the panel Figure4.
As shown in Figure 5 for the entire sample, the IRAC coverage of our samples is not uniform. The median depth of the SZE cluster observations reach[4.5]=21 AB, for instance, while the median depth for the X-ray sample reaches [4.5]=22.5 AB. For the sake of overall consistency of our analysis and to be able to calibrate our method to a depth that the vast majority of current and future Spitzer surveys can easily reach(with even 90 s exposure), we adopt [4.5]cut=21
AB as the magnitude cut for all subsequent analyses. We will also further investigate the dependence of richness estimates on image depth in Section4.1.
For galaxy stellar populations formed at high redshift, a negative k-correction provides a nearly constant 4.5μm flux density over a wide redshift range. AnL[*4.5]galaxy formed at zf=3 will have [4.5]∼21 (AB) at 0.4z2.0, which is
sufficiently bright that it is robustly seen in even just 90 s integrations with Spitzer (e.g., Eisenhardt et al.2008).
While we recognize that using the simple approach of a single apparent magnitude cut at 0.4<z<2.0 would introduce a bias for optical mass–richness relationships, we note here that an infrared relation is not significantly affected because of the k-correction. Adopting Mancone et al. (2010) results on the evolution with redshift of the characteristic absolute magnitude M[* ( )4.5]z , we note that at 4.5μm, in the redshift range spanned by our sample, the stellar population evolution and redshift evolution are roughly matched, thus sampling a similar rest-frame luminosity range of the cluster galaxy population as a function of redshift. Because of cluster galaxy population evolution with redshift seen through the 4.5μm band filter at 0.4z2, our adopted apparent magnitude limit [4.5]cut always corresponds to a roughly
similar absolute magnitude M[4.5]cut~M[*4.5]( )z +1 over this large redshift range. Wefind in fact thatM[4.5]cutvaries between
* ( ) [ ]
M4.5 z + 0.87 (at z 1.2) and M[*4.5]( )z +1.17 (at z ∼ 0.5), thus by 0.3 mag. This small variation in limit magnitude will not significantly increase the scatter in the mass–richness relation we will derive in the next Section5. In Section 4.1, based on a subsample of clusters for which deeper data are available, we will study the dependence of richness estimates on survey depth and will parameterize a linear relation (Equation (3)) to account for these effects. Accordingly, a variation in magnitude cut by 0.3 mag will result in a variation in richness, DR~6 gals ´Mpc-2, hence in logarithmic scale ΔLog R∼0.05, which is very small and will not significantly increase the scatter in the derived mass–richness relation.
4. Derivation ofSpitzer 4.5 μm Richness
The richness of a cluster is a measure of the surface density of galaxies associated with that cluster within a given radius. Because of the presence of background and foreground field galaxies and foreground stars, one cannot identify which source in the vicinity of a cluster belongs to the cluster. Richness is therefore a statistical measure of the galaxy population of a cluster, based on some operational definition of cluster membership and an estimate of foreground/background subtraction. Furthermore, as we aim to provide an efficient and inexpensive 4.5μm photometric proxy of cluster mass within R500, we need to adopt a sufficiently large aperture
radius in which galaxies should be counted in a way that minimizes the Poisson scatter in richness and takes into account the typical R500 of clusters at z>0.4 and the angular size
constraint defined by the single pointing Spitzer field of view. Thus we define a richness parameter, R[4.5], as the background-subtracted projected surface density of sources with[4.5]<21
Figure 4.4.5μm number counts for four representative clusters in our sample (black dashed histogram) compared to number counts of the deeper SpUDS controlfield (red solid histogram). The left column shows the number counts for two X-ray clusters while the right column shows the number counts for two SZE clusters.
Figure 5. Histogram of the 4.5μm depths reached by the archival data available for the X-ray(blue) and the SZE (red) samples. The dashed lines indicate the median depth of each sample.
AB within 2 arcmin from the cluster center, expressed in units of galaxies Mpc−2.
We first measure the number of objects in the vicinity of the cluster, NCluster, with[4.5]<21 mag within 2 arcmin of the cluster
center determined from the SZE or X-ray data(bottom-left panel of Figure6). In order to estimate the number of background sources (stars and galaxies) to subtract, we use the SpUDS survey to derive a mean blank-field surface density of sources above the same magnitude limit. To estimate this, we measure the number of sources above the magnitude limit within an aperture radius of 2 arcmin from each source with [4.5]<21 in the SEIP photometric catalog of the SpUDS field. We then fit a Gaussian to the distribution, iteratively clipping at 2σ (see the bottom-right panel of Figure 6). The resulting mean of the distribution, áNFieldñ =76 gals, is then subtracted from NCluster.
This method of background subtraction assumes that the stellar density in the SpUDSfield is the same as that in the cluster field, which need not be true due to the structure of our galaxy. As we deal with an all-sky archival sample of clusters, we correct for the variation of the foreground star counts with Galactic latitude. Using the Wainscoat et al. (1992) mode predictions7for the IR point-source sky, we can estimate the number of stars with [4.5]<21 within 2 arcmin from the center of each cluster in our sample, NS, and compare it to the average value for the SpUDS
field, NS,Field=9.4. Thus we can correct our richness estimate at
the location of each cluster for the difference in star counts by
Figure 6.Top left panel:[4.5]-band image of a representative cluster in our sample at z=1.132. The white dashed circle indicated has a radius r=2′. Top right panel: positions of all the sources extracted by the photometric pipeline from the 4.5μm band image of the cluster and indicated by black diamonds. Sources with magnitudes[4.5]<21 AB are indicated by open red diamonds. The black circle has a radius r=2′, centered on the reported cluster center. Magnitude-selected sources that are also within the circle are indicated withfilled red symbols. Bottom left panel: [4.5] magnitude distribution of all sources in the Spitzer/IRAC image of this cluster. The red dotted–dashed line indicates the magnitude cut adopted consistently throughout this work. The number of sources, NCluster, brighter than the [4.5]cut=21 AB and within 2′ from the cluster center is indicated. Bottom right panel: distribution of the number of sources in the control field brighter than [4.5]=21 and within r<2′ from each source extracted in the SpUDS photometric catalog. The red line indicates a 2σ clipped Gaussian fit of the distribution. The red dotted–dashed line indicates the mean of the Gaussian fit, áNFieldñ, which is used for the source background correction throughout this work, as described in the
text. Clearly, the clusterfield has more than twice as many objects within the aperture.
7
Web tool available athttp://irsa.ipac.caltech.edu/applications/Background
subtracting the difference between these numbers:
= - á ñ -( - ) ( )
[ ]
R4.5 NCluster NField NS NS,Field , 2 where values NClusterand NSfor each cluster in our sample are
listed in Tables1 and2.
To test thefidelity of the calibrated model of the galaxy adopted here, we have also compared Wainscoat et al. (1992) predictions with the ones from a more recent model of the galaxy, TRILEGAL (Girardi et al. 2012) . At each of the 116 cluster positions, TRILEGAL has been run 10 times with varying input parameters (IMFs, extinction laws, model of the thin/thick disk, halo and bulge model) to output the mean and stdev values for the number of stars within 2 arcmin. Wefind the results of the two models in remarkably good agreement. At the coordinates of our sample clusters, wefind the median difference between the outputs of the two models to be only∼1.5 stars in a 2 arcmin radius. The mean difference of the two models is found to be ∼4.3 stars in a 2 arcmin radius. This difference is two orders of magnitude smaller than the typical total source counts, NCluster, at the location of the
clusters(see Tables1and2) and is hence negligible with respect to the typical errors(Poissonian statistics) reported here.
Finally we normalize for the surface area subtended by the 2′ radius aperture at the redshift of each cluster and express R[4.5] in units of gals Mpc−2throughout the paper (unless specified). We note that since projected areas evolve slowly with redshift, in particular at high redshift, our method is also suitable for clusters for which only a photometric redshift is available. For instance, for a zphot=1.0 cluster, even a large uncertainty in redshift of
Δz=±0.1 would only result in a variation of area of just ∼5%, implying a small variation of the inferred R[4.5].
The derived richness values for our sample of clusters are listed in Tables 1 and 2. Richness uncertainties account for Poisson fluctuations in background counts and cluster counts as well as the uncertainty in the mean background counts shown in Figure6.
We note that we do not adopt a color criterion in our richness definition. The [3.6]–[4.5] color is known to be degenerate with redshift at z 1.3, but can be used as an effective redshift indicator (e.g., Papovich2008; Muzzin et al.2013; Wylezalek et al.2013; Rettura et al.2014) at z>1.3. The method takes advantage of the fact that the [3.6]–[4.5] color is a linear function of redshift between 1.3z1.5 and at z1.5 the color reaches a plateau out to z∼3 (see also left panel of Figure7). While an IRAC
color cut [3.6]–[4.5]>−0.1 (AB) is effective at identifying galaxies at z > 1.3, due to the color degeneracy at lower redshifts, having at least one shallow optical band in addition, would be required for alleviating contamination from foreground interlopers at z<0.3 (see discussion in Muzzin et al. 2013). Since optical data are unavailable for the large part of our archival sample and>90% of our sample is comprised of cluster galaxies at z<1.3 we do not include a color cut in our definition of richness. We also note that by measuring richness at 4.5μm, corresponding to rest-frame near-infrared bands at the redshifts spanned by our sample, we are tracing the masses of galaxies better than optical richness estimates because stellar mass-to-light ratios show less scatter in the NIR than in the optical(e.g., Bell & de Jong2001).
We remind that our method is based on counts in cells centered on either the X-ray or the SZE central positions as reported in the literature. As current and future Spitzer-selected cluster survey may adopt our method to estimate a mass proxy as well, it is valuable to attempt to estimate how using instead Spitzer-determined cluster centers would affect the richness estimation, hence the derived cluster mass estimates.
To test the effect of miscentering on richness estimates we have blindly determined cluster centers from Spitzer data directly in afield where samples of confirmed spectroscopic clusters had also published X-ray derived centers. In our test we implemented the similar clusterfinding and centering algorithm described in Rettura et al.(2014) and ran a cluster search on the Spitzer data of the Bootesfield (SDWFS; Ashby et al.2009).
We identify five infrared selected clusters associated to spectroscopically confirmed clusters at 1.3<z<1.75, where X-ray data are also available in the literature (Brodwin et al. 2011, 2013, 2016). We measure the mean positional offset, Δpos, of the newly derived Spitzer centers with respect to the
published center coordinates. We find Dpos= ¢ ¢0. 2 0. 1. We then shift the centers of all the clusters in this paper in a random direction byΔposand derive newR[4.5]shift values. We find the
mean inferred number of objects in the vicinity of the shifted cluster position, Nclustershift, to vary by 8% with respect to the previous Ncluster estimates reported in Tables 1 and 2. Using
Equation(2), this translates to a difference in R[4.5]on the order of 0.06 dex, which is more than a factor of four smaller than our estimated uncertainties in mass atfixed richness.
Figure 7.Evolution of the[3.6]–[4.5] color (left panel) and H−[4.5] color (middle panel) with redshift for a set of Bruzual & Charlot (2003) stellar population
models with exponentially declining star-formation rates withτ=0.1 Gyr (early-type galaxy) and τ=1.0 Gyr (star forming galaxy). These colors are used to translate our measure of[4.5] μm richness into a H-band richness estimate. The right panel shows the predicted richness (in gals arcmin−2) for Euclid clusters at 0.4<z<2.0, in the wide-area survey (Hcut= 24 AB) as a function of cluster mass.
4.1. Dependence of Richness on Survey Depth and Aperture Radius
In order to investigate the effect of the chosen aperture radius and depth of the IRAC 4.5μm data on richness estimates, we performed a series of tests on a subsample of clusters where the IRAC data is deep enough to allow us to measure richness values at different sensitivity levels. As shown in Figure5, the
X-ray sample contains a “deep” subsample of 36 clusters for which their depth is 22.5 mag AB. We measure the average (and standard deviation of the mean) richness of this sample down to various depths, [4.5]cut, ranging 21<[4.5]<22.5,
and with different aperture radii,0. 5¢ < < ¢r 2 . As shown in Figure 8, richness increases with increasing magnitude cut adopted; this is not surprising since there are typically more
Table 1
X-Ray Selected Cluster Sample
Clus ID R.A. Decl. NAME z log M500,X log R[4.5] NCluster NS
(deg., J2000) (deg., J2000) (Me) (galaxies Mpc−2)
26 4.6408 16.4381 MACS J0018.5+1626 0.5456 14.995±0.035 1.645-+0.0710.061 162 14.32 46 7.64 26.3044 WARP J0030.5+2618 0.5 14.506±0.029 1.716-+0.0650.056 173 19.31 51 8.9971 85.2214 WARP J0035.9+8513 0.8317 14.462±0.034 1.721-+0.0640.056 242 37.17 145 25.3846 −30.5783 400d J0141-3034 0.442 14.514±0.028 1.604-+0.0740.064 134 8.74 156 28.1721 −13.9703 WARP J0152.7-1357 0.833 14.638±0.035 1.448-+0.0910.075 149 8.60 187 34.1404 −17.7908 WARP J0216.5-1747 0.578 14.435±0.030 1.119-+0.1400.106 101 8.75 200 37.6108 18.6061 400d J0230+1836 0.799 14.671±0.035 1.520-+0.0830.070 167 15.70 268 52.1504 −21.6678 400d J0328-2140 0.59 14.545±0.031 1.819-+0.0570.050 208 10.59 276 53.2925 −24.9447 400d J0333-2456 0.475 14.477±0.029 1.459-+0.0890.074 123 10.73 312 58.9971 −37.6961 400d J0355-3741 0.473 14.528±0.029 1.683-+0.0670.058 155 12.39 316 61.3512 −41.0042 400d J0405-4100 0.686 14.524±0.032 1.525-+0.0820.069 156 13.23 355 73.5462 −3.015 MACS J0454.1-0300 0.5377 14.954±0.034 1.676-+0.0680.059 178 25.66 380 80.2937 −25.51 400d J0521-2530 0.581 14.491±0.030 1.357-+0.1020.083 135 23.86 382 80.5575 −36.4136 400d J0522-3624 0.472 14.412±0.028 1.589-+0.0760.065 150 22.29 405 85.7117 −41.0014 RDCS J0542-4100 0.642 14.585±0.032 1.552-+0.0800.067 170 26.83 550 132.1983 44.9392 RX J0848.7+4456 0.574 14.100±0.028 1.375-+0.1000.081 126 13.56 551 132.2346 44.8711 RX J0848.9+4452 1.261 14.328±0.041 0.889-+0.1930.133 105 13.54 557 133.3058 57.9956 400d J0853+5759 0.475 14.435±0.028 1.683-+0.0670.058 157 14.01 586 141.6521 12.7164 400d J0926+1242 0.489 14.405±0.028 1.567-+0.0780.066 141 14.00 601 145.7796 46.9975 RXC J0943.1+4659 0.4069 14.679±0.030 1.939-+0.0490.044 192 10.53 621 149.0121 41.1189 400d J0956+4107 0.587 14.465±0.030 1.322-+0.1070.086 118 9.88 631 150.5321 68.98 400d J1002+6858 0.5 14.455±0.029 1.568-+0.0780.066 142 13.35 634 150.7671 32.9078 400d J1003+3253 0.4161 14.711±0.030 1.830-+0.0560.050 168 9.63 713 164.2479 −3.6244 MS1054.4-0321 0.8309 14.661±0.035 1.454-+0.0900.075 154 12.61 743 169.375 17.7458 400d J1117+1744 0.547 14.417±0.029 1.525-+0.0820.069 137 8.73 747 170.0321 43.3019 WARP J1120.1+4318 0.6 14.634±0.032 1.547-+0.0800.068 146 8.31 748 170.2429 23.4428 400d J1120+2326 0.562 14.522±0.030 1.647-+0.0710.061 159 8.37 825 180.5571 57.8647 400d J1202+5751 0.677 14.508±0.032 1.372-+0.1000.081 129 9.45 864 185.3542 49.3019 400d J1221+4918 0.7 14.605±0.033 1.580-+0.0770.065 163 8.55 865 185.5079 27.1553 400d J1222+2709 0.472 14.417±0.028 1.522-+0.0830.069 127 8.11 873 186.74 33.5472 WARP J1226.9+3332 0.888 14.779±0.038 1.281-+0.1130.089 127 8.02 971 198.0808 39.0161 400d J1312+3900 0.404 14.426±0.027 1.688-+0.0670.058 139 8.48 1020 203.585 50.5181 ZwCl 1332.8+5043 0.62 14.530±0.031 1.682-+0.0680.058 176 9.29 1050 206.875 −11.7489 RXC J1347.5-1144 0.4516 15.221±0.037 1.726-+0.0640.056 162 15.81 1063 208.57 −2.3628 400d J1354-0221 0.546 14.418±0.029 1.687-+0.0670.058 169 12.94 1066 209.3308 62.545 400d J1357+6232 0.525 14.474±0.029 1.635-+0.0720.061 154 11.18 1089 213.7962 36.2008 WARP J1415.1+3612 0.7 14.473±0.032 1.498-+0.0850.071 149 9.61 1094 214.1171 44.7772 NSCS J141623+444558 0.4 14.531±0.028 1.778-+0.0600.053 154 9.77 1107 215.9492 24.0781 MACS J1423.8+2404 0.543 14.909±0.034 1.642-+0.0710.061 157 10.25 1171 229.4829 31.4597 WARP J1517.9+3127 0.744 14.332±0.032 1.033-+0.1580.115 105 12.11 1184 231.1679 9.9597 WARP J1524.6+0957 0.516 14.576±0.030 1.522-+0.0830.069 140 15.69 1264 250.4679 40.0247 400d J1641+4001 0.464 14.520±0.029 1.564-+0.0780.066 142 18.84 1410 275.4087 68.4644 RX J1821.6+6827 0.8156 14.453±0.034 1.135-+0.1370.104 132 29.93 1506 309.6225 −1.4214 RX J2038.4-0125 0.673 14.373±0.031 1.208-+0.1240.096 165 62.22 1519 314.0908 −4.6308 MS2053.7-0449 0.583 14.425±0.030 1.799-+0.0580.052 231 40.92 1548 322.3579 −7.6917 MACS J2129.4-0741 0.594 14.873±0.034 1.652-+0.0700.060 181 24.79 1658 345.7004 8.7306 WARP J2302.8+0843 0.722 14.377±0.031 1.158-+0.1330.102 117 16.19
Table 2
SZE-selected Cluster Sample
Clus ID R.A. Decl. NAME z log M500,SZ log R[4.5] NCluster NS
(deg., J2000) (deg., J2000) (Me) (galaxies Mpc−2)
OBJ1 3.05417 16.0375 MOO J0012+1602 (1) 0.944 14.146-+0.0850.071 1.505-+0.0820.069 172 14.39 OBJ4 49.8517 −0.4225 MOO J0319-0025 1.194 14.491-+0.0290.027 0.902-+0.1780.126 104 12.19 OBJ5 153.535004 0.64056 MOO J1014+0038 1.27 14.531-+0.0260.025 1.423-+0.0910.075 165 13.49 OBJ7 228.677917 13.77528 MOO J1514+1346 1.059 14.342-+0.0640.055 1.501-+0.0830.069 176 14.02 OBJ8 216.637299 35.139889 IDCS J1426.5+3508 (2) 1.75 14.415-+0.0630.055 0.990-+0.1660.120 109 9.94 OBJ10 34.432999 −3.76 XLSSU J021744.1-034536(3) 1.91 14.127-+0.0180.017 0.973-+0.1710.122 107 9.52 OBJ9 86.655128 −53.757099 SPT-CL J0546-5345(4) 1.067 14.703-+0.0370.034 1.346-+0.0910.075 161 27.47 OBJ11 310.248322 −44.860229 SPT-CL J2040-4451 1.478 14.522-+0.0450.041 1.395-+0.0870.072 177 28.46 OBJ12 31.442823 −58.48521 SPT-CL J0205-5829 1.322 14.675-+0.0370.034 1.194-+0.1220.095 130 12.67 OBJ13 316.52063 −58.745075 SPT-CL J2106-5844 1.132 14.922-+0.0330.031 1.418-+0.0860.072 172 24.45 OBJ16 355.299103 −51.328072 SPT-CL J2341-5119 1.003 14.747-+0.0360.033 0.950-+0.1660.120 105 12.16 OBJ17 93.964989 −57.776272 SPT-CL J0615-5746 0.972 15.023-+0.0330.031 1.410-+0.0810.068 176 35.12 OBJ18 326.64624 −46.550034 SPT-CL J2146-4633 0.933 14.737-+0.0380.035 1.228-+0.1100.088 132 17.61 OBJ20 83.400879 −50.09008 SPT-CL J0533-5005 0.881 14.578-+0.0440.040 0.802-+0.1790.126 100 24.49 OBJ21 15.729427 −49.26107 SPT-CL J0102-4915 0.8701 15.159-+0.0330.030 1.496-+0.0850.071 162 10.60 OBJ22 9.175811 −44.184902 SPT-CL J0036-4411 0.869 14.512-+0.0520.047 1.414-+0.0940.077 147 10.13 OBJ23 72.27417 −49.024605 SPT-CL J0449-4901 0.792 14.69-+0.0390.036 1.384-+0.0920.076 146 17.81 OBJ24 359.922974 −50.164902 SPT-CL J2359-5009 0.775 14.557-+0.0450.041 1.013-+0.1540.114 104 11.54 OBJ25 353.105713 −53.967545 SPT-CL J2332-5358 0.402 14.723-+0.0390.036 1.291-+0.1030.083 105 12.88 OBJ26 325.139099 −57.457577 SPT-CL J2140-5727 0.4054 14.531-+0.0540.048 1.476-+0.0770.065 126 20.02 OBJ27 69.574867 −54.321243 SPT-CL J0438-5419 0.4214 15.033-+0.0340.031 1.582-+0.0710.061 137 17.77 OBJ28 87.904144 −57.155659 SPT-CL J0551-5709 0.423 14.696-+0.0410.037 1.626-+0.0620.054 154 28.82 OBJ29 62.815441 −48.321751 SPT-CL J0411-4819 0.4235 14.913-+0.0350.032 1.387-+0.0910.075 115 14.52 OBJ30 323.916351 −57.44091 SPT-CL J2135-5726 0.427 14.789-+0.0370.034 1.638-+0.0650.057 148 20.49 OBJ31 321.146179 −61.410179 SPT-CL J2124-6124 0.435 14.715-+0.0400.037 1.453-+0.0770.065 130 22.74 OBJ32 52.728668 −52.469772 SPT-CL J0330-5228 0.4417 14.824-+0.0360.034 1.570-+0.0750.064 134 13.24 OBJ33 77.337387 −53.705322 SPT-CL J0509-5342 0.4607 14.704-+0.0400.037 1.028-+0.1160.091 104 21.00 OBJ34 60.968086 −57.323669 SPT-CL J0403-5719 0.4664 14.574-+0.0490.044 1.475-+0.0810.069 129 15.99 OBJ35 103.962601 −52.567741 SPT-CL J0655-5234 0.4703 14.707-+0.0420.038 1.351-+0.0650.056 158 56.19 OBJ36 326.468201 −56.747559 SPT-CL J2145-5644 0.48 14.840-+0.0360.033 1.688-+0.0630.055 164 19.35 OBJ37 308.801147 −52.851883 SPT-CL J2035-5251 0.5279 14.793-+0.0380.035 1.741-+0.0570.050 194 29.60 OBJ38 354.352264 −59.704929 SPT-CL J2337-5942 0.775 14.926-+0.0340.031 1.402-+0.0920.076 144 14.14 OBJ39 82.019592 −53.002384 SPT-CL J0528-5300 0.7678 14.562-+0.0460.041 1.273-+0.0980.080 137 23.77 OBJ40 345.466888 −55.776756 SPT-CL J2301-5546 0.748 14.429-+0.0600.052 1.090-+0.1330.102 111 14.34 OBJ41 31.279436 −64.545746 SPT-CL J0205-6432 0.744 14.532-+0.0500.045 1.303-+0.1030.083 130 14.62 OBJ42 310.8284 −50.593838 SPT-CL J2043-5035 0.7234 14.656-+0.0430.040 1.474-+0.0770.066 165 27.70 OBJ43 314.217407 −54.993736 SPT-CL J2056-5459 0.718 14.545-+0.0480.043 1.491-+0.0770.065 165 25.30 OBJ44 315.093262 −45.805138 SPT-CL J2100-4548 0.7121 14.466-+0.0660.057 1.340-+0.0910.075 142 24.02 OBJ45 47.629108 −46.783417 SPT-CL J0310-4647 0.7093 14.635-+0.0440.040 1.091-+0.1400.105 107 11.51 OBJ46 19.598965 −51.943447 SPT-CL J0118-5156 0.705 14.575-+0.0490.044 1.449-+0.0900.074 143 11.00 OBJ47 0.249912 −57.806423 SPT-CL J0000-5748 0.7019 14.659-+0.0400.037 1.378-+0.0960.078 135 13.07 OBJ48 68.254105 −56.502499 SPT-CL J0433-5630 0.692 14.496-+0.0560.050 1.080-+0.1270.098 112 17.78 OBJ49 80.301186 −51.076565 SPT-CL J0521-5104 0.6755 14.614-+0.0430.039 1.494-+0.0780.066 159 22.39 OBJ50 38.255245 −58.327393 SPT-CL J0233-5819 0.663 14.594-+0.0460.041 1.391-+0.0940.077 134 13.00 OBJ51 33.106094 −46.950199 SPT-CL J0212-4657 0.6553 14.770-+0.0380.034 1.473-+0.0870.073 142 10.42 OBJ52 335.712189 −48.573456 SPT-CL J2222-4834 0.6521 14.734-+0.0390.036 1.477-+0.0840.070 147 15.01 OBJ53 77.920914 −51.904373 SPT-CL J0511-5154 0.645 14.611-+0.0440.040 1.376-+0.0890.074 139 21.06 OBJ54 85.716667 −41.004444 SPT-CL J0542-4100 0.642 14.713-+0.0410.038 1.405-+0.0820.069 148 26.83 OBJ55 40.861546 −59.512436 SPT-CL J0243-5930 0.6352 14.661-+0.0420.038 1.427-+0.0900.074 137 13.56 OBJ56 319.731659 −50.932484 SPT-CL J2118-5055 0.6254 14.557-+0.0530.047 1.301-+0.0950.078 130 21.38 OBJ57 326.531036 −48.780003 SPT-CL J2146-4846 0.623 14.592-+0.0490.044 1.585-+0.0720.062 165 17.95 OBJ58 89.925095 −52.826031 SPT-CL J0559-5249 0.609 14.762-+0.0370.034 1.349-+0.0830.070 143 30.61 OBJ59 314.587891 −56.14529 SPT-CL J2058-5608 0.606 14.468-+0.0600.053 1.294-+0.0920.076 132 25.23 OBJ60 326.69574 −57.614769 SPT-CL J2146-5736 0.6022 14.570-+0.0470.043 1.417-+0.0860.072 139 19.48 OBJ61 356.184692 −42.720924 SPT-CL J2344-4243 0.596 15.081-+0.0330.031 1.625-+0.0720.062 162 10.91 OBJ62 64.345047 −47.813923 SPT-CL J0417-4748 0.581 14.870-+0.0350.033 1.258-+0.1070.086 117 14.87 OBJ63 83.608215 −59.625652 SPT-CL J0534-5937 0.5761 14.439-+0.0640.055 1.223-+0.0960.079 125 25.93
galaxies at fainter luminosities for a canonical luminosity function. This test validates the importance of adopting a uniform magnitude cut while dealing with a heterogeneous, archival sample. We also note that the slope and standard deviation of richness is much smaller for the larger, adopted 2 arcmin radius aperture than for the smaller apertures. For the adopted 2 arcmin aperture radius, the dependence of the mean richness, áR[4.5]ñ, with the magnitude cut adopted, [4.5]cut, is
bestfitted with the linear relation
á ñ = - + ´ - ( ) ( ) [ ] ( ) [ ] R galaxies Mpc 358.71 18.97 4.5 AB mag. 3 4.5 2 cut
This relation allows us to quantify the expected increase in average richness value at increasing depths of the observa-tions due simply to an intrinsic photometric effect, not due to
cluster-to-cluster variations or dynamical state. By means of extrapolation, this relation could be also used to predict the expected average richness value for samples of clusters at similar redshifts with upcoming, wide-area, infrared surveys (see discussion in Section6.4).
5. Calibrating a Cluster Mass–Richness Relation at 0.4<z<2.0
In this section, we calibrate mass–richness relations based on our richness estimates defined in Section4and on total cluster masses as described in Sections 2.1and 2.2. In Figure9, we show the relations we find for the 47 clusters of the X-ray sample (top panel) and the 69 clusters of the SZE sample (bottom panel). We perform a weighted linear least squares fit of the data for each sample individually with a single linear relation on log quantities, where errors in both variables are also taken into account (Press et al. 2002, Section 15.6), andfind = + - ( ) ( ) · [ ] ( ) M M R log 13.68 0.17 0.57 0.23 log gals Mpc , 4 500,X 4.5 2 and = + - ( ) ( ) · [ ] ( ) M M R log 13.34 0.21 0.93 0.22 log gals Mpc . 5 500,SZ 4.5 2
In Figure 10, we show the relation we find for the 116 clusters of the combined sample(solid black line):
= + - ( ) ( ) · [ ] ( ) M M R log 13.56 0.25 0.74 0.18 log gals Mpc . 6 500 4.5 2
To test the robustness of ourfit, we have also run a bootstrap Monte Carlo test, in which the mass–richness relation is repeatedly resampled to reveal whether or not a small sample of
Table 2 (Continued)
Clus ID R.A. Decl. NAME z log M500,SZ log R[4.5] NCluster NS
(deg., J2000) (deg., J2000) (Me) (galaxies Mpc−2)
OBJ64 352.960846 −50.863926 SPT-CL J2331-5051 0.576 14.748-+0.0370.034 1.256-+0.1110.088 114 12.34 OBJ65 327.181213 −61.277969 SPT-CL J2148-6116 0.571 14.649-+0.0430.039 1.473-+0.0800.067 144 20.27 OBJ66 74.116264 −51.27684 SPT-CL J0456-5116 0.5615 14.707-+0.0400.036 1.450-+0.0830.069 139 19.00 OBJ67 38.187614 −52.957821 SPT-CL J0232-5257 0.5559 14.729-+0.0400.036 1.432-+0.0900.075 129 11.75 OBJ68 305.027344 −63.243397 SPT-CL J2020-6314 0.5361 14.515-+0.0540.048 1.306-+0.0820.069 137 33.81 OBJ69 304.483551 −62.978218 SPT-CL J2017-6258 0.5346 14.587-+0.0470.042 1.359-+0.0780.066 142 34.27 OBJ70 346.729767 −65.091042 SPT-CL J2306-6505 0.5298 14.758-+0.0390.036 1.742-+0.0600.053 182 16.94 OBJ71 56.724724 −54.650532 SPT-CL J0346-5439 0.5297 14.738-+0.0390.036 1.541-+0.0770.066 143 14.41 26 4.640833 16.438056 MACS J0018.5+1626 (5) 0.5456 14.938-+0.0440.040 1.645-+0.0710.061 162 14.31 355 73.54625 −3.015 MACS J0454.1-0300(5) 0.5377 14.858-+0.0610.054 1.676-+0.0680.059 178 25.66 621 149.012083 41.118889 400d J0956+4107 (5) 0.587 14.844-+0.0550.049 1.322-+0.1070.086 118 9.89 1050 206.875 −11.748889 RXC J1347.5-1144(5) 0.4516 15.026-+0.0310.029 1.726-+0.0640.056 162 15.81
Note.M500,SZvalues as reported by(1) Brodwin et al. (2015), (2) Brodwin et al. (2012), (3) Mantz et al. (2014), (4) Bleem et al. (2015), (5) Planck Collaboration
et al.(2015).
Figure 8. Dependence of average richness, áR[4.5]ñ, with adopted 4.5μm
magnitude cuts, [4.5]cut, and aperture radii for the reference X-ray “deep sample.” Error bars indicate the standard deviation of the mean.
clusters could, for instance, dramatically alter the result of the fit. We run the least-square fitting algorithm 1000 times and at each repetition we randomly toss out 25% of the sample. We then infer the mean and standard deviation of the intercept and slope distribution for the 1000 fit results. We find the latter values in perfect agreement with the ones of Equation(6).
We have also checked whether a small subsample of high-mass clusters could largely alter the result of thefit. However, even excluding the three most massive clusters in the sample, the resulting values of intercept and slope and their errors are still consistent within 1σ of the ones presented in Equation (6).
Based on the 68.3% confidence regions of the fits (dotted lines), we estimate the associated errors in mass at fixed richness to be ±0.25 dex. We will discuss the dependence of the scatter of this relation with concentration in Section 6.2. The intrinsic scatter of the relation is measured in the R[4.5] direction around the best-fitting R[4.5]–M relation for that sample via bootstrapping method (Tremaine et al. 2002; Kelly 2007; Andreon & Hurn 2013) and is denoted as sR[4.5]∣M. Wefind sR[4.5]∣M=0.32dex for our sample.
We compare the measured scatter in our relation with literature richness and mass estimates. Using an r-band luminosity-based optical richness estimator, RL; Planck
Colla-boration et al.(2014) found the associated error in mass at fixed richness to be±0.27 dex. The intrinsic scatter of their RL–M
relation, sR ML∣ =0.35, is also similar to the value that we have found. Note that RL was defined by Wen et al. (2012) for a
large sample of low-redshift Sloan Digital Sky Survey(SDSS, York et al.2000) selected clusters for which X-ray masses were provided by Piffaretti et al.(2011). We note that their method cannot be extended to all clusters in our sample because SDSS lacks coverage of the southern sky and because deeper optical data than the one available from SDSS would be required to detect the bulk of cluster galaxies at 0.6<z2.
Also at lower redshifts, 0.03<z<0.55, Andreon & Hurn (2010) and Andreon (2015) used multiband SDSS photometry to define an optical richness estimator, n200, aimed at counting
red cluster members within a specified luminosity range and color as a proxy of the total mass, M200, within the R200radius.
These predicted masses are found to have a smaller 0.16 dex scatter with mass, but require optical photometry in at least two bands.
As a comparison with other observable-mass scaling relations, we note that Maughan (2007) studied the LX–M relation for
Figure 9.4.5μm richness–mass relation for a sample of 47 X-ray selected clusters (top panel) and 69 SZE-selected clusters (bottom panel) at 0.4<z<2.0. The dashed lines correspond to the best straight-line fits to data with errors in both coordinates for each sample respectively. The dotted lines indicate the 68.3% confidence regions of each fit.
Figure 10.4.5μm richness–mass relation for a sample of 47 X-ray selected clusters (blue circles) and 69 SZE-selected clusters (red circles) at 0.4<z<2.0. The blue and red dashed lines correspond to the best straight-linefits to the individual samples shown previously in Figure9. The solid line indicates thefit to the combined sample and the dotted lines indicate the 68.3% confidence regions of this fit.
a sample of 115 clusters at 0<z<1.3 and found the associated error in mass atfixed luminosity to be ±0.21 dex. They measured the intrinsic scatter in the LX–M relation to be sL M∣ =0.39when all of the core emission is included in their LX measurements.
Interestingly, they have also demonstrated that the scatter can greatly be reduced to sL M∣ =0.17by excising the core emission in their LX measurements. Furthermore, Rozo et al. (2014a,
2014b) used SZE data from Planck and X-ray data from Chandra and XMM-Newton to calibrate YX–M, YSZ–YX and YSZ–M relations for different samples of clusters at z<0.3. They found low values for the scatters of the YSZ–M relations, sYSZ∣M=
– 0.12 0.20.
To summarize, the method we have proposed here requires only shallow, single pointing, single band observations and an estimate of the cluster center position and redshift to provide reliable richness-based cluster mass estimates. The very low observational cost associated with our approach makes it potentially available for very large samples of clusters at 0.4<z<2.0.
6. Discussion
In this section, we discuss the richness distribution and galaxy surface density profile for our samples. We also examine potential sources of the scatter in the mass–richness relation, including sample selection and galaxy concentration. We then explore the possibility of extending our method to other MIR all-sky surveys like WISE, and the implications of ourfindings on future, wide-field near-infrared cluster surveys like Euclid.
6.1. Richness Distribution and Profiles
In the top panel of Figure 11 we show the distributions of richness for our samples. The X-ray sample (blue histogram and blue dashed line) shows larger median richness values than the SZ sample (red histogram and red dashed line). This difference is statistically significant, as shown by the Kolmogorov–Smirnov (KS) test, which provides a probability PKS∼10−6 that the observed distributions of richness are
extracted from the same parent population.
As the SZE sample probes the redshift range z>1 more extensively, we tested whether this difference in median richness could depend on the relative redshift distribution of the two samples. Hence we performed the KS test again only for clusters in the redshift range 0.4<z<0.8, which is well sampled by both methods. However, the latter test produces a similar outcome of the former, ruling out this possibility.
In the bottom panel of Figure11we show the dependence of richness on aperture radius for both cluster samples. We divide each sample in two equally populated bins of X-ray or SZE derived cluster masses. We calculate the richness profiles by measuring the background-subtracted projected surface density of galaxies with[4.5]<21 AB within r=30″, 60″, 120″ from the cluster center. Wefind the shapes of the richness profiles to be similar for both samples and in each mass bins and the richness values consistent within the large scatter. However, the low-mass X-ray sample appears to be as rich as the high-mass SZ sample. Hence at fixed cluster mass, X-ray clusters appear to have, on average, higher near-infrared richness than SZ-selected clusters.
One possibility is that this difference is due to systematics in derived masses for the X-ray and SZ samples. For example, if, despite the careful re-calibration in Section2, the X-ray masses are underestimates, then the higher richness of the X-ray clusters would imply that they are more massive clusters. Given the extensive work on calibrating X-ray and SZ observables with mass, this seems unlikely. Yet, without having a statistically significant SZ, X-ray, and richness estimates sample of the same clusters in this redshift range, we cannot definitively rule it out.
Alternately, could this be due to a selection effect, where the X-ray surveys, at a fixed cluster mass, typically select richer systems where galaxy merging has been, on average, less effective than in SZE-selected clusters? X-ray surveys are indeed usually considered biased toward selecting more relaxed, more evolved systems(e.g., Eckert et al.2011). This is because of the presence of a surface brightness peak in the so-called“cool core” clusters. The clear peak of X-ray emission
Figure 11.Top: histogram of richness of the X-ray(blue) and the SZE (red) samples. The dashed lines indicate the median richness of each sample. Bottom: dependence of richness with cluster mass and aperture radius for the X-ray(blue) and the SZE (red) samples.
is more easily detected in the wide shallow surveys of ROSAT, for instance, and it is considered to be associated with a decrease in the gas temperature, typical of relaxed structures.
On the other hand, most of the clusters newly discovered by Planck via SZE show clear indication of morphological disturbances in their X-ray images, suggesting a more active dynamical state (Planck Collaboration et al. 2011). Interest-ingly, Rossetti et al. (2016), measuring the offset between the X-ray peak and the BCG population, a known indicator of an active cluster dynamical state (e.g., Sanderson et al. 2009; Mann & Ebeling2012), found evidence of the dynamical state of SZ-selected clusters to be significantly different from X-ray-selected samples, with a higher fraction of non-relaxed, merging systems.
In the hierarchical cluster formation scenario, clusters form by the infall of less massive groups along thefilaments. Therefore while it is possible that a larger fraction of SZ-selected clusters in our sample are in a less developed stage of cluster formation, if the difference in richness between our samples was to be explained with the fact that X-ray clusters were more evolved and had accreted more galaxies within R500, then they should
also be the most massive. This has, however, not been found, as clearly shown in Figure 3, unless there are mass calibration uncertainties larger than those discussed.
An intriguing possibility is that there are differences in the intrinsic baryon fraction within R500 relative to the
cosmolo-gical baryon fraction between the cluster samples. If X-ray clusters at these redshifts harbor a larger baryon fraction per unit dark-matter halo mass compared to SZE cluster samples within the aperture radius, it could account for lower total masses, more efficient cooling of the intracluster medium by thermal Bremsstrahlung, and the resultant increased star-formation efficiency could result in a larger population of luminous galaxies translating to a higher richness. Indeed, simulations like the millennium simulations show a factor of two variation in the baryon fraction of >1014M☉dark-matter halos, which could easily account for both the differences in derived masses and richness values. Even among the X-ray cluster sample, Vikhlinin et al.(2009) showed that the baryon fraction increases with increasing mass among which there is likely the origin of a richness–mass correlation that we derive. A comparison between the density profiles of SZ clusters and X-ray clusters in Planck Collaboration et al.(2011) shows that SZ clusters show shallower density profiles than X-ray-selected clusters, which may again argue that the baryons in SZE-selected clusters are predominantly at larger radii than in the X-ray sample. However, extracting effects such as these from the data would again require SZ, X-ray, and richness estimates of the same clusters in this redshift range while our study has rather heterogeneous samples with different origins that we have attempted to place on the same calibration scale. Apart from stating these possibilities, it is challenging for us to definitively claim one of them as an origin for the observed difference.
6.2. Galaxy Concentration
In an attempt to understand the source of the scatter in the mass–richness relation, we also measure the galaxy concentra-tion of our cluster samples, defined as the ratio between the richness measured within r=60″ and r=120″. By definition, a higher value of galaxy concentration corresponds to systems with a steeper galaxy surface density profile. As shown in
Figure12, there is a hint that clusters that deviate the most from thefitted mass–richness relation are also the ones with the most centrally concentrated galaxy surface density profile. If we include a correction for galaxy concentration, the scatter of the mass–richness relation slightly decreases so that the associated error in mass at fixed richness is found to be ±0.22 dex, indicating that surface density concentration does play a significant role in the scatter.
Possible origins for this are blending and source confusion in the IRAC image, beam dilution when the 2′ aperture is much larger than the cluster overdensity, or the impact of galaxy merging on richness estimates. Images of galaxy clusters that are more centrally concentrated are more likely to be affected by the blending of some cluster galaxies in the core, given the Spitzer angular resolution. This could result in underestimating their richness. The amount of confusion is linearly proportional to the surface density of galaxies. A cluster that is four times as concentrated in surface density would have its richness underestimated by 0.6 dex, which is the amount of offset of the red points in Figure12from the best-fit line.
Alternatively, systems with higher central galaxy concentra-tions have their richness estimate, calculated within a radius r=120″, biased by the fact that the aperture chosen is too large, hence their richness is relatively smaller compared to less centrally concentrated systems. However, if this was purely an observational bias, we should have found a correlation between galaxy concentration and θ500 derived for the hot gas in the
intracluster medium, a proxy of cluster size, which is not apparent.
If mergers were responsible for the scatter in the richness– mass relation, clusters of the same total mass that might have experienced more or less merger events in their cores with respect to their outskirts would result in lower or higher concentration measurements and would therefore be found to have lower or higher values of richness. In the most extreme cases, as shown in Figure13, galaxy clusters of the same total mass(as probed by their gas) and at the same lookback time may have experienced very different evolutionary processes,
Figure 12.4.5μm richness vs. mass relation color-coded by galaxy concentration for the combined sample. The solid line indicates the linearfit to the sample where errors in both variables are taken into account. The dotted lines indicate the 68.3% confidence regions of this fit.
resulting in large differences in richness and concentration inferred from the number and location of their cluster members. Thus we conclude that the scatter we derive in the richness– mass relation is likely due to a combination of source confusion and differences in evolutionary history of the clusters.
6.3. WISE 4.5μm Richness
The AllWISE8program combined data from the cryogenic Wide-field Infrared Survey Explorer mission (WISE, Wright et al. 2010) and the NEOWISE (Mainzer et al. 2011) post-cryogenic survey to deliver a survey of the full MIR sky. Since all-sky catalogs in the W2 band at 4.6μm are available, it could allow us to potentially extend our method outside the Spitzer footprint. Therefore in this section we test whether our proposed method of deriving MIR richness estimates can be applied robustly to the publicly available WISE data.
To this aim, we use archival WISE 4.6μm data available at the location of all our clusters and apply the same method described in Section 4. The SEIP source list contains W2 photometry for positional counterparts found in the AllWISE release both for all clusters in our sample and for the control SpUDS field. In Figure 14(left panel), we show the richness estimates based on data from Spitzer and WISE for our SZE sample. The dashed lines indicates the straight-line fit to be compared to the 1:1(solid) line. We note that there are several catastrophic outliers and that Spitzer-based richness values are systematically higher than the WISE-based counterparts. We note that WISE is less sensitive than Spitzer and that its angular resolution is also poorer (6 4 versus 2 0).
To test whether the catastrophic discrepancies in the Spitzer versus WISE richness estimates could be ascribed to confusion, we sum theflux densities of every object detected within r=120″ from each cluster center by the two instruments. We also subtract a median flux density from the backgroundfield to get a flux overdensity at the location of the cluster. Source confusion makes groups of sources in a high resolution image appear as a single bright source in a low-resolution image. By adding the flux densities we take out the effect of confusion, which would bias low richness estimates. We then use the redshift of the cluster to translate the summedflux overdensity to a luminosity surface density. As shown in the right panel of Figure 14, we find that the total luminosity densities for each cluster appear to be conserved, as the two instruments provide matching mea-surements. Therefore we can ascribe the aforementioned discrepancies solely to the poorer angular resolution of WISE, with richness estimates highly depending on the particular projected cluster galaxy geometry. At the WISE image quality, we expect a higher number of sources to be blended, resulting in lower counts of galaxies per cluster, yielding lower richness values than those measured by Spitzer. For example, source overdensities of 30 galaxies in the 2′ radius aperture would correspond to 10 gals Mpc−2, which in turn would correspond to a ratio of 11 for the number of WISE beams per source. This is well below the classical confusion limit. In reality, the confusion is even higher since the average underlying foreground source density would also contribute to confusion noise.
To summarize, we deem WISE-based richness estimates to be poorer proxies for cluster mass preventing us from effectively extending our method beyond the Spitzer footprint. Calibrating a mass–richness relation for the WISE data set will require a different technique and is therefore beyond the scope of this paper.
6.4. Future Wide-field Near-infrared Cluster Surveys The upcoming Euclid and WFIRST missions aim to survey large portions of the extragalactic sky in the near infrared(e.g., H band) to measure the effects of dark energy, but also have distant cluster studies as a key scientific goal. The Euclid wide-area survey, in particular, will observe 15,000 deg2, almost the entire extragalactic celestial sphere, down to a 5σ point-source depth of H=24 mag (AB). They plan to use photometric redshift overdensities to identify clusters but that requires ancillary ground-based optical data which is currently being taken. In this section, based on the results of our analysis, we try to provide a simple prediction of the expected richness values for Euclid-selected clusters and the range of masses that will be accessible.
Euclid is expected to detect ∼2×106 clusters at all redshifts, with ∼4×104 of them at 1<z<2 with cluster masses M2008×10
13
Me(Sartoris et al.2016; Ascaso et al. 2017). The cluster sample in our study spans a similar mass and redshift range, hence we can attempt to predict the average richness expected for clusters at 0.4<z<2.0 as a function of mass in the Euclid survey.
In Figure7, we show the evolution of the[3.6]–[4.5] (left panel) and H−[4.5] color (middle panel) with redshift for a set of Bruzual & Charlot (2003) stellar population models with exponentially declining star-formation rates. We show both the typical color evolution expected for an early-type galaxy (assuming τ = 0.1 Gyr) and a star-forming galaxy
Figure 13. Spitzer/IRAC images of four clusters in our sample with similar M500,SZbut with different richness or redshifts measured. The white dashed circles have a radius r=2′, centered on the reported cluster. Images (A) and (B) show clusters at similar redshift, z∼0.5, while (C) and (D) are instead at z∼1. Despite having similar redshifts and masses, (A) and (B) are found with large differences in their richness values suggesting that there may be other dependencies. Images(A) and (C), however, show clusters of the same mass, found with a similar richness despite being at different redshift.
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